Any sore that will not heal should be checked out by your primary care doctor
. There are some times of non healing sores that occur on the breast that can be serious. This is especially the case if you have a sore on the nipple, as this may, rarely, be a sign of Paget's disease, which is a type of breast cancer. Much more common causes of nipple sores would include fungal infections and sores simply from irritation from rubbing on clothing. However, it is worth having checked out just to make sure.
Sores on the skin of the breast away from the nipple may also occur for various reasons. These include again irritation and chafing from clothing, as well as cysts or local infections that may be chronically inflamed or draining and need to be looked at. If you have sores or areas of redness especially underneath of the breasts where they touch your skin, then this is most likely to be the result of a fungal infection, which is very common in this location.
The bottom line is that you should set up a visit with your doctor so that they can diagnosis this and decide if treatment is necessary.